Problem: The geometric sequence $a_i$ is defined by the formula: $a_1 = -4$ $a_i = a_{i - 1} \cdot 2$ Find the sum of the first $50$ terms in the sequence. Choose 1 answer: Choose 1 answer: (Choice A) A $-5.07\cdot10^{29}$ (Choice B) B $ -4.50\cdot10^{15} $ (Choice C) C $-2.25\cdot10^{15}$ (Choice D) D $-1.50\cdot10^{15}$
Getting started Let's write out the first few terms of the series: $-4 -8 -16...$ We're dealing with a geometric series because each term is multiplied by $2$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-4})$ and the number of terms $(n = {50})$ are given in the question. The common ratio $r$ is ${2}$ because each term is multiplied by ${2}$ to get the next term. Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{50}}&=\dfrac{{-4}(1-\left({2}\right)^{{50}})}{1-\left({2}\right)} \\\\ S_{{50}}&=4(1-\left({2}\right)^{{50}}) \\\\ S_{{{50}}} &\approx -4.50\cdot10^{15} \end{aligned}$ The answer $ -4.50\cdot10^{15} $